A047055 -id:A047055 - cp's OEIS frontend

A052562 a(n) = 5^n * n!.

1, 5, 50, 750, 15000, 375000, 11250000, 393750000, 15750000000, 708750000000, 35437500000000, 1949062500000000, 116943750000000000, 7601343750000000000, 532094062500000000000, 39907054687500000000000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

A simple regular expression in a labeled universe.

For n >= 1 a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_5)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 504
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000142, A008548, A008546, A034325, A000165, A047055, A047056, A051150, A092514, A092618.

[5^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 05 2011

spec := [S,{S=Sequence(Union(Z,Z,Z,Z,Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combstruct):A:=[N,{N=Cycle(Union(Z$5))},labeled]: seq(count(A,size=n)/5,n=1..16); # Zerinvary Lajos, Dec 05 2007

Table[5^n*n!, {n, 0, 20}] (* Wesley Ivan Hurt, Sep 28 2013 *)

{a(n) = 5^n*n!}; \\ G. C. Greubel, May 05 2019

[5^n*factorial(n) for n in (0..20)] # G. C. Greubel, May 05 2019

a(n) = A051150(n+1, 0) (first column of triangle).
E.g.f.: 1/(1-5*x).
a(n) = 5*n*a(n-1) with a(0)=1.
G.f.: 1/(1-5*x/(1-5*x/(1-10*x/(1-10*x/(1-15*x/(1-15*x/(1-20*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
G.f.: 1/Q(0), where Q(k) = 1 - 5*x*(2*k+1) - 25*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013
a(n) = n!*A000351(n). - R. J. Mathar, Aug 21 2014
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/5) (A092514).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/5) (A092618). (End)

Name changed by Arkadiusz Wesolowski, Oct 04 2011

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

Original entry on oeis.org

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512

Offset: 0

Dale Gerdemann, Apr 12 2015

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:
  a\b 1.......2.......3.......4.......5.......6
  -1  A144431
  0   A007318 A038208 A038221
  1   A008292 A256890 A257180 A257606 A257607
  2   A060187 A257609 A257611 A257613 A257615
  3   A142458 A257610 A257620 A257622 A257624 A257626
  4   A142459 A257612 A257621
  5   A142460 A257614 A257623
  6   A142461 A257616 A257625
  7   A142462 A257617 A257627
  8   A167884 A257618
  9   A257608 A257619
The row sums of these, and similarly constructed number triangles, are shown in the following table:
  a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
  0   A000079 A000302 A000400
  1   A000142 A001715 A001725 A049388 A049198
  2   A000165 A002866 A002866 A051580 A051582
  3   A008544 A051578 A037559 A051605 A051607 A051609
  4   A001813 A047053 A000407 A034177 A051618 A051620 A051622
  5   A047055 A008546 A008548 A034300 A034325 A051688 A051690
  6   A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
  7   A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
  8   A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
  9   A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
  10                                  A051262 A035265 A035273         A035277
  11                                  A254322
  12                                          A145448
The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).
  a\b  0           1           2          3
  -2   A130595/1
  -1
  0
  1    A110555/-1  A120434/-1  A144697/1  A144699/2
With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).
If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

			Array, t(n, k), begins as:
   1,    2,      4,        8,        16,         32,          64, ...;
   2,   12,     52,      196,       684,       2276,        7340, ...;
   4,   52,    416,     2644,     14680,      74652,      357328, ...;
   8,  196,   2644,    26440,    220280,    1623964,    10978444, ...;
  16,  684,  14680,   220280,   2643360,   27227908,   251195000, ...;
  32, 2276,  74652,  1623964,  27227908,  381190712,  4677894984, ...;
  64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
    1;
    2,     2;
    4,    12,      4;
    8,    52,     52,       8;
   16,   196,    416,     196,      16;
   32,   684,   2644,    2644,     684,      32;
   64,  2276,  14680,   26440,   14680,    2276,     64;
  128,  7340,  74652,  220280,  220280,   74652,   7340,   128;
  256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172,   256;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.

Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.

Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.

A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022

Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)

t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015

def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (End)

A085157 Quintuple factorials, 5-factorials, n!!!!!, n!5.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 14, 24, 36, 50, 66, 168, 312, 504, 750, 1056, 2856, 5616, 9576, 15000, 22176, 62832, 129168, 229824, 375000, 576576, 1696464, 3616704, 6664896, 11250000, 17873856, 54286848, 119351232, 226606464, 393750000, 643458816

Offset: 0

Hugo Pfoertner, Jun 21 2003

nonn

The term "Quintuple factorial numbers" is also used for the sequences A008546, A008548, A052562, A047055, A047056 which have a different definition. The definition given here is the one commonly used.

			a(12) = 168 because 12*a(12-5) = 12*a(7) = 12*14 = 168.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Multifactorials
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. n!:A000142, n!!:A006882, n!!!:A007661, n!!!!:A007662, n!!!!!!:A085158, 5-factorial primes: n!!!!!+1:A085148, n!!!!!-1:A085149.

Cf. A288092.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-5);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 18 2019

b:= func< n | (n lt 6) select n else n*Self(n-5) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 18 2019

a:= n-> `if`(n < 1, 1, n*a(n-5)) end proc; seq(a(n), n = 0..40); # G. C. Greubel, Aug 18 2019

a[n_]:= If[n<1, 1, n*a[n-5]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Aug 18 2019 *)
Table[Times@@Range[n,1,-5],{n,0,40}] (* Harvey P. Dale, May 12 2020 *)

a(n)=if(n<1, 1, n*a(n-5))
for(n=0,50,print1(a(n),",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 19 2006

def A085157(n):
    if n <= 0:
        return 1
    else:
        return n*A085157(n-5)
n = 0
while n <= 40:
    print(n,A085157(n))
    n = n+1 # A.H.M. Smeets, Aug 18 2019

def a(n):
    if (n<1): return 1
    else: return n*a(n-5)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 18 2019

a(n) = 1 for n < 1, otherwise a(n) = n*a(n-5).

Sum_{n>=0} 1/a(n) = A288092. - Amiram Eldar, Nov 10 2020

A084947 a(n) = Product_{i=0..n-1} (7*i+2).

Original entry on oeis.org

1, 2, 18, 288, 6624, 198720, 7352640, 323516160, 16499324160, 956960801280, 62202452083200, 4478576549990400, 353807547449241600, 30427449080634777600, 2829752764499034316800, 282975276449903431680000, 30278354580139667189760000, 3451732422135922059632640000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Harvey P. Dale, Table of n, a(n) for n = 0..340

Cf. A000079, A000142, A000165, A008544, A001813, A045754, A047055, A047657, A084942, A084947, A084948, A084949, A144739, A144827, A049209, A051188.

List([0..20], n-> Product([0..n-1], k-> 7*k+2) ); # G. C. Greubel, Aug 18 2019

[ 1 ] cat [ &*[ (7*k+2): k in [0..n-1] ]: n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008

a := n->product(7*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Join[{1},FoldList[Times,7*Range[0,15]+2]] (* Harvey P. Dale, Nov 27 2015 *)
Table[7^n*Pochhammer[2/7, n], {n,0,15}] (* G. C. Greubel, Aug 18 2019 *)

vector(20, n, n--; prod(k=0, n-1, 7*k+2)) \\ G. C. Greubel, Aug 18 2019

[product(7*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

a(n) = A084942(n)/A000142(n)*A000079(n) = 7^n*Pochhammer(2/7, n) = 7^n*Gamma(n+2/7)/Gamma(2/7).
D-finite with recurrence a(0) = 1; a(n) = (7*n - 5)*a(n-1) for n > 0. - Klaus Brockhaus, Nov 10 2008
G.f.: 1/(1-2*x/(1-7*x/(1-9*x/(1-14*x/(1-16*x/(1-21*x/(1-23*x/(1-28*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-5)^n*Sum_{k=0..n} (7/5)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 7*x)^(2/7).
a(n) ~ sqrt(2*Pi)*7^n*n^n/(exp(n)*n^(3/14)*Gamma(2/7)). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/7^5)^(1/7)*(Gamma(2/7) - Gamma(2/7, 1/7)). - Amiram Eldar, Dec 19 2022

a(15) from Klaus Brockhaus, Nov 10 2008

A084940 Heptagorials: n-th polygorial for k=7.

Original entry on oeis.org

1, 1, 7, 126, 4284, 235620, 19085220, 2137544640, 316356606720, 59791398670080, 14050978687468800, 4018579904616076800, 1374354327378698265600, 553864793933615401036800, 259762588354865623086259200, 140271797711627436466579968000, 86407427390362500863413260288000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A006472, A001044, A000680, A084939, A084941, A084942, A084943, A084944, A085356, A246745.

a := n->n!/2^n*mul(5*i+2,i=0..n-1); [seq(a(j),j=0..30)];

polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[ polygorial[7, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)
Join[{1},FoldList[Times,PolygonalNumber[7,Range[20]]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 29 2019 *)

a(n)=n!/2^n*prod(i=1,n,5*i-3) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = polygorial(n, 7) = (A000142(n)/A000079(n))*A047055(n) = (n!/2^n)*Product_{i=0..n-1}(5*i+2) = (n!/2^n)*5^n*Pochhammer(2/5, n) = (n!/2^n)*5^n*Gamma(n+2/5)*sin(2*Pi/5)*Gamma(3/5)/Pi.
D-finite with recurrence 2*a(n) = n*(5*n-3)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 5^n * n^(2*n + 2/5) * Pi /(Gamma(2/5) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A047657 Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+2).

Original entry on oeis.org

1, 2, 16, 224, 4480, 116480, 3727360, 141639680, 6232145920, 311607296000, 17450008576000, 1081900531712000, 73569236156416000, 5444123475574784000, 435529878045982720000, 37455569511954513920000, 3445912395099815280640000, 337699414719781897502720000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..340

Cf. A007559, A008542, A011781, A073005.

Cf. A000165, A008544, A001813, A047055, A084947, A084948, A084949.

List([0..20], n-> Product([0..n-1], k-> 6*k+2) ); # G. C. Greubel, Aug 18 2019

[1] cat [(&*[6*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019

a:= n->product(6*j+2, j=0..n-1); seq(a(n), n=0..20); # G. C. Greubel, Aug 18 2019

b[1]=2; b[n_]:= b[n] = b[n-1] +6; a[0]=1; a[1]=2; a[n_]:= a[n] = a[n-1]*b[n]; Table[a[n], {n,0,20}] (* Roger L. Bagula, Sep 17 2008 *)
FoldList[Times,1,6*Range[0,20]+2] (* Harvey P. Dale, Aug 06 2013 *)
Table[6^n*Pochhammer[1/3, n], {n,0,20}] (* G. C. Greubel, Aug 18 2019 *)

vector(20, n, n--; prod(k=0, n-1, 6*k+2)) \\ G. C. Greubel, Aug 18 2019

[product(6*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

E.g.f.: (1-6*x)^(-1/3).
a(n) = 2^n*A007559(n).
a(n) = A084941(n)/A000142(n)*A000079(n) = 6^n*Pochhammer(1/3, n) = 1/2*6^n*Gamma(n+1/3)*sqrt(3)*Gamma(2/3)/Pi. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
Let b(n) = b(n-1) + 6; then a(n) = b(n)*a(n-1). - Roger L. Bagula, Sep 17 2008
G.f.: 1/(1-2*x/(1-6*x/(1-8*x/(1-12*x/(1-14*x/(1-18*x/(1-20*x/(1-24*x/(1-26*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-4)^n*Sum_{k=0..n} (3/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/G(0) where G(k) = 1 - x*(6*k+2)/( 1 - 6*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
D-finite with recurrence: a(n) +2*(-3*n+2)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*(Gamma(1/3) - Gamma(1/3, 1/6))/6^(2/3). - Amiram Eldar, Dec 18 2022
a(n) ~ sqrt(Pi) * 2^(n+1/2) * (3/e)^n * n^(n-1/6) / Gamma(1/3). - Amiram Eldar, Sep 01 2025

A047056 Quintuple factorial numbers: Product_{k=0..n-1} (5*k+3).

Original entry on oeis.org

1, 3, 24, 312, 5616, 129168, 3616704, 119351232, 4535346816, 195019913088, 9360955828224, 496130658895872, 28775578215960576, 1812861427605516288, 123274577077175107584, 8999044126633782853632, 701925441877435062583296, 58259811675827110194413568

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A008546, A008548, A047055, A048994, A052562.

List([0..20], n-> Product([0..n-1], k-> 5*k+3 )); # G. C. Greubel, Aug 20 2019

[1] cat [(&*[5*k+3: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 20 2019

seq(mul(5*k+3, k = 0..n-1), n = 0..20); # G. C. Greubel, Aug 20 2019

Table[5^n*Pochhammer[3/5, n], {n,0,20}] (* G. C. Greubel, Aug 20 2019 *)
Join[{1},FoldList[Times,5*Range[0,20]+3]] (* Harvey P. Dale, Oct 08 2020 *)

vector(20, n, n--; prod(j=0,n-1, 5*j+3) ) \\ G. C. Greubel, Aug 20 2019

[5^n*rising_factorial(3/5, n) for n in (0..20)] # G. C. Greubel, Aug 20 2019

E.g.f.: (1-5*x)^(-3/5).
a(n) ~ sqrt(2*Pi)/Gamma(3/5)*n^(1/10)*(5*n/e)^n*(1 - 11/300*n^ - 1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
G.f.: 1/(1-3*x/(1-5*x/(1-8*x/(1-10*x/(1-13*x/(1-15*x/(1-18*x/(1-20*x/(1- ... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-2)^n*Sum_{k=0..n} (5/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/G(0)  where G(k) = 1 - x*(5*k+3)/( 1 - 5*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
G.f.: G(0)/2, where G(k)= 1  + 1/(1 - (5*k+3)*x/((5*k+3)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
D-finite with recurrence: a(n) +(-5*n+2)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/5^2)^(1/5)*(Gamma(3/5) - Gamma(3/5, 1/5)). - Amiram Eldar, Dec 19 2022

A114799 Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 18, 30, 44, 60, 78, 98, 120, 288, 510, 792, 1140, 1560, 2058, 2640, 6624, 12240, 19800, 29640, 42120, 57624, 76560, 198720, 379440, 633600, 978120, 1432080, 2016840, 2756160, 7352640, 14418720, 24710400, 39124800

Offset: 0

Jonathan Vos Post, Feb 18 2006

Many of the terms yield multifactorial primes a(n) + 1, e.g.: a(2) + 1 = 3, a(4) + 1 = 5, a(6) + 1 = 7, a(9) + 1 = 19, a(10) + 1 = 31, a(12) + 1 = 61, a(13) + 1 = 79, a(24) + 1 = 12241, a(25) + 1 = 19801, a(26) + 1 = 29641, a(29) + 1 = 76561, a(31) + 1 = 379441, a(35) + 1 = 2016841, a(36) + 1 = 2756161, ...

Equivalently, product of all positive integers <= n congruent to n (mod 7). - M. F. Hasler, Feb 23 2018

			a(40) = 40 * a(40-7) = 40 * a(33) = 40 * (33*a(26)) = 40 * 33 * (26*a(19)) = 40 * 33 * 26 * (19*a(12)) = 40 * 33 * 26 * 19 * (12*a(5)) = 40 * 33 * 26 * 19 * 12 5 = 39124800.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.
Index entries for sequences related to factorial numbers

Cf. A045754, A084947, A288094.

Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A085158 (and A008542, A047058, A047657), A045755.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-7);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 20 2019

b:= func< n | (n lt 8) select n else n*Self(n-7) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 20 2019

A114799 := proc(n)
    option remember;
    if n < 1 then
        1;
    else
        n*procname(n-7) ;
    end if;
end proc:
seq(A114799(n),n=0..40) ; # R. J. Mathar, Jun 23 2014
A114799 := n -> product(n-7*k,k=0..(n-1)/7); # M. F. Hasler, Feb 23 2018

a[n_]:= If[n<1, 1, n*a[n-7]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Aug 20 2019 *)

A114799(n,k=7)=prod(j=0,(n-1)\k,n-j*k) \\ M. F. Hasler, Feb 23 2018

def a(n):
    if (n<1): return 1
    else: return n*a(n-7)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 20 2019

a(n) = 1 for n <= 1, else a(n) = n*a(n-7).

Sum_{n>=0} 1/a(n) = A288094. - Amiram Eldar, Nov 10 2020

Edited by M. F. Hasler, Feb 23 2018

A084948 a(n) = Product_{i=0..n-1} (8*i+2).

Original entry on oeis.org

1, 2, 20, 360, 9360, 318240, 13366080, 668304000, 38761632000, 2558267712000, 189311810688000, 15523568476416000, 1397121162877440000, 136917873961989120000, 14513294639970846720000, 1654515588956676526080000, 201850901852714536181760000, 26240617240852889703628800000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

G. C. Greubel, Table of n, a(n) for n = 0..330

Cf. A000079, A000142, A000165, A008544, A001813, A047055, A047657, A048994, A084943, A084947, A084949.

List([0..20], n-> Product([0..n-1], k-> 8*k+2) ); # G. C. Greubel, Aug 18 2019

[1] cat [(&*[8*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019

a := n->product(8*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Table[8^n*Pochhammer[1/4, n], {n,0,20}] (* G. C. Greubel, Aug 18 2019 *)

vector(20, n, n--; prod(k=0, n-1, 8*k+2)) \\ G. C. Greubel, Aug 18 2019

[product(8*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

a(n) = A084943(n)/A000142(n)*A000079(n) = 8^n*Pochhammer(1/4, n) = 1/2*Gamma(n+1/4)*sqrt(2)*Gamma(3/4)*8^n/Pi.
a(n) = (-6)^n*Sum_{k=0..n} (4/3)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+2)/(2*x*(8*k+2) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 8*x)^(1/4).
a(n) ~ sqrt(2*Pi)*8^n*n^n/(exp(n)*n^(1/4)*Gamma(1/4)). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/8^6)^(1/8)*(Gamma(1/4) - Gamma(1/4, 1/8)). - Amiram Eldar, Dec 20 2022

A084949 a(n) = Product_{i=0..n-1} (9*i+2).

Original entry on oeis.org

1, 2, 22, 440, 12760, 484880, 22789360, 1276204160, 82953270400, 6138542009600, 509498986796800, 46873906785305600, 4734264585315865600, 520769104384745216000, 61971523421784680704000, 7932354997988439130112000, 1086732634724416160825344000, 158662964669764759480500224000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Robert Israel, Table of n, a(n) for n = 0..329

Cf. A000165, A008544, A001813, A047055, A047657, A084947, A084948.

Cf. A000079, A000142, A035012, A048994, A084944.

List([0..20], n-> Product([0..n-1], k-> 9*k+2) ); # G. C. Greubel, Aug 19 2019

[1] cat [(&*[9*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 19 2019

a:= n-> product(9*i+2,i=0..n-1); seq(a(j),j=0..20);

Table[9^n*Pochhammer[2/9, n], {n,0,20}] (* G. C. Greubel, Aug 19 2019 *)

vector(20, n, n--; prod(k=0, n-1, 9*k+2)) \\ G. C. Greubel, Aug 19 2019

[product(9*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 19 2019

a(n) = A084944(n)/A000142(n)*A000079(n) = 9^n*Pochhammer(2/9, n) = 9^n*Gamma(n+2/9)/Gamma(2/9).
a(n) = (-7)^n*Sum_{k=0..n} (9/7)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
E.g.f.: (1-9*x)^(-2/9). - Robert Israel, Mar 22 2017
D-finite with recurrence: a(n) + (-9*n+7)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1 + (e/9^7)^(1/9)*(Gamma(2/9) - Gamma(2/9, 1/9)). - Amiram Eldar, Dec 21 2022
a(n) ~ sqrt(2*Pi) * (9/e)^n * n^(n-5/18) / Gamma(2/9). - Amiram Eldar, Aug 30 2025

cp's OEIS Frontend

A052562 a(n) = 5^n * n!.

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

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A085157 Quintuple factorials, 5-factorials, n!!!!!, n!5.

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A084947 a(n) = Product_{i=0..n-1} (7*i+2).

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A084940 Heptagorials: n-th polygorial for k=7.

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A047657 Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+2).

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.